Stereographic protractor.



Patented- Feb. 5, M.

-s; L. PENFIELD. v STEREOGRAPHIC PBOTRACTOR.

(Application filed Oct. 25, 1900.)

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No. 667,570. Patented Feb. 5, 19m.

8. L. PENFIELD. STEREOGBAPHIC PROTBACTOB.

(Application filed Oct 25, 1900.

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IUNIITED STATES rion.

ATENT STEREOG RAPHIC PROTRACTOR.

SPECIFICATION forming part of Letters Patent No. 667,570, dated February5, 1901.

Application filed October 25, 1900. serial No. 34,248. We model.)

To aZZ whom it may concern.-

Be it known that I, SAMUEL LEWIS PEN- FIELD, of New Haven, in the countyof New Haven and State of Connecticut, have invented a new Improvementin Stereographic Protractors; and I do hereby declare the following,when taken in connection with the accompanying drawings and the lettersof reference marked thereon, to be a full, clear, and exact descriptionof the same, and which said drawings constitute part of thisspecification, and represent, in

Figure 1, a plan view of one form which my improved stereographicprotractor may assume; Fig. 2, a graduated circle drawn to the samescale and illustrating such a circle as may be used in conjunction withmy protractor.

Myinvention relates to an improved stereographic protractor, the objectbeing to produce an extremely simple, cheap, and convenient instrumentfor the ready solution of problems in spherical trigonometry as they maybe encountered in the study or teaching 2 fiof that particular branch ofmathematics or as applied to special subjectssuch as crystallography,astronomy, navigation, &c.to which the principles of sphericaltrigonometry are applied.

My improved device is particularly adapted for use in high schools andcolleges, Where practical application of the uses of sphericaltrigonometry are made in the course of instruction of students inscience.

With these ends in view my invention consists in a stereographicprotractor consisting of a transparent or semitransparent plate or sheethaving imposed upon it a series of stereographically-projected arcs ofcircles.

My invention further consists in a stereographic protractor havingcertain other details, as will be hereinafter described, and pointed outin the claims.

Before proceeding to a description of the instrument it may be statedthat in a stereographic projection any given point or line on thesurface of a sphere is represented as projected upon a iiat surfacepassing through the center of the sphere, the said surface forming theplane of projection. The lines of projection all run to a point or poleon the surface of the sphere, the said pole being at right angles to theplane of projection.

Myimproved instrument in the form chosen for illustration consists ofarectangular plate or sheet A, of transparent celluloid or othertransparent or semitransparent material, upon which certain lines andfigures are imposed, either by printing, engraving, or in any othersuitable manner. The flat surface of this plate represents the plane ofprojection which is to be conceived as passing through the center of asphere of certain diameter, which in this case is fourteen centimeters.

In further explanation of what I mean by plane of projection let it beimagined that we have a sphere of fourteen centimeters in diameter, andtherefore corresponding to the diameter of the graduated circle B ofmyimproved stereographic protractor. Let it be further imagined thatthis supposed sphere has an equator on it and that the equator isdivided into degrees. Now a plane passing through the said equator andwhich may eX- tend beyond the sphere will be the plane of projectionWithin the meaning of that term as I have used it. The pole or pointtoward which everythingis projected will be thesouth pole of the sphere,provided the plane of projection is the plane of the equator. The saidlines upon the plate A comprise the graduated circle B, (correspondingto the equator of a fourteen-centimeter sphere,) divided into degreesrunning from zero to one hundred and eighty and numbered from 10 to 170in opposite directions for convenience of reading, though it would besufficient to number them in one direction only. Upon the circle B, Iimpose a line C, joining the zero and onehundred-and-eighty degreepoints upon the graduated circle and having its ends slightly enlarged,as at c c, for convenience in setting the zero andone-hundred-and-eighty degree points of the graduation at any desiredfixed points. This line C, it may be stated, representes astereographically-projected great circle. I also impose upon the circlea line D, joining the ninety-degree points upon the graduated circle andof course centrally intersecting the line C, already mentioned. Thisline D also represents a stereographically-projected great circle. Ialso impose upon the circle a progressive series ofstereographically-projected arcs of small circles E ofconstantly-increasing radii and secured by calculations based upon knownprinciples of stereographic projection not requiring rehearsal herebecause well known to those familiar with the subject of sphericaltrigonometry. One half of the graduated circle is divided into spaces often degrees by such arcs of circles drawn in full lines and into spacesof five degrees by such arcs of circles drawn in broken lines, while theother half of the circle is divided into spaces of ten degrees by sucharcs of circles drawn in full lines and into spaces of one degree byalternating arcs of circles drawn in light full lines and light brokenlines. This method of indicating even degrees by full lines and odddegrees by broken lines has been adopted merely as a matter ofconvenience, for within a fou rteen-centimeter circle the lines arenecessarily crowded together, and it makes the use of the protractorsomewhat simpler to have them thus diiferentiated. The calculationsemployed for securing the said arcs of circles take into account thediameter of the graduated circle, which, as aforesaid, is four-.

teen centimeters. It follows from this that my improved stereographicprotractor can only be used in connection with some scheme of projectionfounded upon a corresponding scale. Inother words, a protractor the graduated circle of which has a diameter of fourteen centimeters can only beemployed in conjunction with a stereographic projection made upon acircle of fourteen centimeters diameter. If I wish to employ a larger ora smaller scale for the projection, it would be necessary to construct anew protractor corresponding to the diameter of the circle adopted.

An engraved circle of fourteen centimeters diameter intended to be usedin connection with my stereographic protractor, Fig. l, is shown by Fig.2 of the drawings. This circle may be printed upon sheets of papersupplied for use in conju nction with my improved protractor, or it maybe drawn by the use of the ordinary tools.

The practical use of my improved instrument will be snfficientlydemonstrated by a single problem in navigation. This problem may bestated as follows: Required the distance in degrees between Queenstown,latitude fifty-one degrees fifty minutes north, longitude eight degreesfifteen minutes west, and New York, latitude forty degrees forty minutesnorth,longitude seventy-four degrees west. This problem may be solved bythe use of my protractor in different ways; but perhaps the simplestprocedure for the solu tion of it is as follows: The plane of theprojection is chosen as a north and south plane, corresponding to thenorth and south meridian passing through Queenstown, this plane beingeight degrees and fifteen minutes west of the meridian of Greenwich.Queenstown is easily located at Q, fifty degrees fifty minutes north ofeast, which represents the intersection of the meridian with theequator. The circular are N P S on Fig. 2 is astereographically-projected north and south meridian passing through NewYork, seventy-four degrees west of Greenwich, and hence sixtyfivedegrees forty-five minutes west of the meridian of Queenstown. Themeridian N P S is easily plotted by means of suitable scales,

and the exact location of New York-forty degrees forty-one minutesnorthis easily determined. It is not necessary to draw the arc Q Q,which represents a great circle passing through Queenstown and New Yorkand terminating at Q opposite Queenstown. New

by applying my protractor so that its zeroand one-hundred-and-eightydegree points correspond to Q (Queenstown) and Q the distance in degreesbetween Queenstown and New York can be determined directly as a littleover forty-five degrees. By actual measurement with my protractor it wasfound to be forty-five degrees fifteen minutes. By

calculation the distance is forty-five degrees eleven minutes. Inmeasuring this distance with my protractor it will be understood, ofcourse, that after the points Queenstown and New York are located on thegraduated circle shown by Fig. 2 of the drawings the protractor is laidover the said circle, so as to register therewith, the zero points beingat Q and Q, after which the solution of the problem may be made byfollowing the stereographically-projected arcs of circles on my improvedinstrument in their relations to the points Queenstown and New Yorkestablished on the circle of Fig. 2. It will be seen from thisillustration that the distance in degrees between any two points of astereographic projection can be measured directly and without anelaborate series of computations by my improved instrument, providedthat the stereographic projection is plotted on a scale corresponding tothe diameter of the protractor.

I do not limit myself toany set size for the protractor nor to minordetails as to just how it is to be printed or engraved. It is preferablyprinted on transparent celluloid or on semitransparent material, such astracing cloth or paper. Itis also apparent that while a protractorgiving a full circle maybe more convenient a semicircular protractorwould answer every requirement. I would therefore have it understoodthat I do not limit myself to the exact details herein shown anddescribed, but hold myself at liberty to make such alterations as fairlyfall within the spirit and scope of my invention.

Having fully described my invention, what I claim as new, and desire tosecure by Letters Patent, is-

1. As a new article of manufacture,astereographic protractor for use inthe solution of problems in spherical trigonometry, the said articleconsisting of a transparent or semitransparent plate having imposed uponit a progressive series of stereographically-projected arcs of circles,whereby the plate may be superposed upon another surface and its imposedlines read in conjunction therewith.

2. As a new article of manufacture,a stereographic protractor for use inthe solution of problems in spherical trigonometry, the said articleconsisting of a transparent or semitransparent plate having imposed uponit a progressive series of stereographically-pro jected arcs of smallcircles intersecting a stereographically-projected arc of a greatcircle, whereby the plate may be superposed upon another surface and itsimposed lines read in conjunction therewith.

3. As a new article of manufacture, a stereographic protractorconsisting of a transparent or semitransparent plate or sheet providedwith a graduated circle or a part of such a circle, a line imposed uponthe circle and joining the zero and one-hundred-andeighty degree pointsthereof, a line imposed upon the circle at a right angle to the saidzero to one-hundred-and-eighty degree line, the center of which itintersects, and a series of stereographically-projected arcs of circlesimposed upon the graduated circle, and intersecting the said zero toone-hundred-andeighty degree line.

In testimony whereof I have signed this specification in the presence oftwo-subscribing witnesses.

SAMUEL LEWIS PENFIELD.

Witnesses:

FREDERIO G. EARLE, GEORGE D. SEYMOUR.

